Proof Definition and 999 Threads

The given definition of a linear transformation ##F## being symmetric on an inner product space ##V## is ##\langle F(\textbf{u}), \textbf{v} \rangle = \langle \textbf{u}, F(\textbf{v}) \rangle## where ##\textbf{u},\textbf{v}\in V##. In the attached image, second equation, how is the...

Summary: Given three points on a positive definite quadratic line, I need to prove that the middle point is never higher than at least one of the other two. I am struggling to write a proof down for something. It's obvious when looking at it graphically, but I don't know how to write the...

I have been struggling with this problem and also my friends. We are not the best at epsilon-delta proof and we have not found an understandable solution to this problem.

Use the formal definition of Big-Oh to prove that if f (n) and g(n) are nonnegative functions such that f (n) = O(g(n)), f (n) + g(n) = Ω(g(n)). By the definition of Big-Oh: If f(n) and g(n) are non-negative functions such that f(n) = O(g(n)) there must be positive constants c and n0 such...

Hey guys, I'm kind of in a rush because I'll have to go to my classes soon here at USF Tampa, but I had one last problem for Intermediate Analysis that needs assistance. Thank you in advance to anyone providing it. Question being asked: "Let $A$ be a nonempty set of real numbers which is...

Hey guys, I have an Intermediate Analysis problem that needs assistance. I've really been having a hard time with it. This is what the question says: "Can it happen that A⊂B (A is a subset of B) and A≠B (A does not equal B), yet sup A=sup B (the supremum of A equals the supremum of B)? If so...

Hi, Can some body please explain me the following question: Use induction on ##n## to show that ##|t^n| = n |t| ## for all strings ##t## and all ##n## . Any idea how to that. I know we have a base case and an induction case but what would be the base case and what would be the induction case...

Homework Statement: Suppose f(n) is a function that accepts an integer n as a parameter. When called with n = 1, it executes 2 instructions. When called with a larger value of n, it executes 10n + 30 instructions, two of which are f(n/2). Prove that f(n) executes 10n lg n + 32n − 30...

So I know the formal definition of Big-O, which states that ##f(n) = O(g(n))## if and only if there exists ##{C > 0, n_0 > 0}## such that ##|f(n)|\leq{C}{g(n)}~\forall{n>n_0}##. Here's what I think the proof should go (please bear with me, I have no idea what I'm doing): Suppose there exists a...

I mean I know they are linear since they obey the ohms law. But I don't quite understand the reasoning that since, say, V=Ldi/dt and taking a derivative is a linear operation therefore it is a linear device?? I can verify that sin'(x) = cos(x) or sin(x+90) so the signal is time shifted but...

Suppose a wave function is a linear combination of 2 stationary states: ##\psi(x)=c_1\psi_1(x)+c_2\psi_2(x)##. By [5.52] and [5.53], we have ##\psi(x+a)=e^{iK_1a}c_1\psi_1(x)+e^{iK_2a}c_2\psi_2(x)##. But to prove [5.49], we need ##K_1=K_2##. That means all the eigenvalues of the "displacement"...

I've been going through Bernard Schutz's A First Course in General Relativity, and I'm hung up on his "proof" of the invariance of the interval. At the beginning of section 1.6, he claims that he will prove the invariance of the interval, and after a few lines shows that the universality of the...

The attempt at a solution ## {\log_{2021}(tan\ 1^\circ)} +{\log_{2021}(tan\ 2^\circ)} +{\log_{2021}(tan\ 3^\circ)} +...+{\log_{2021}(tan\ 89^\circ)} = 0 \\ Antilog_{2021}[{\log_{2021}(tan\ 1^\circ)} +{\log_{2021}(tan\ 2^\circ)} +{\log_{2021}(tan\ 3^\circ)} +...+{\log_{2021}(tan\ 89^\circ)} ]...

Dear all, I am trying to prove a simple thing, that if AxA = BxB then A=B. The intuition is clear to me. If a pair (x,y) belongs to AxA it means that x is in A and y is in A. If a pair (x,y) belongs to BxB it means that x is in B and y is in B. If the sets of all pairs are equal, it means...

Please prove the rules of significant figures. I do not know why when multiplying and dividing we have to retain the same number of significant figures as in the number with the least of them.

Let ##\varepsilon > 0## be arbitrary. Now define ##\delta = \text{min}\{\frac{a}{2}, \varepsilon \sqrt{a}\}##. Now since ##a>0##, we can deduce that ##\delta > 0##. Now assume the following $$ 0< |x-a| < \delta $$ From this, it follows that ##0 < |x-a| < \frac{a}{2} ## and ##0 < |x-a| <...

In many books on number theory they define the well ordering principle (WOP) as: Every non- empty subset of positive integers has a least element. Then they use this in the proof of the division algorithm by constructing non-negative integers and applying WOP to this construction. Is it...

In many books on number theory they define the well ordering principle (WOP) as: Every non- empty subset of positive integers has a least element. Then they use this in the proof of the division algorithm by constructing non-negative integers and applying WOP to this construction. Is it possible...

Let A ⊆ R, let f : A → R, and suppose that (a,∞) ⊆ A for some a ∈ R. Then the following statements are equivalent: i) limx→∞ f(x) = L ii) For every sequence (xn) in A ∩ (a,∞) such that lim(xn) = ∞, the sequence (f(xn)) converges to L. Not even sure how to begin this one, other than the fact...

a) Kepler's first law states that a planet like Earth displays an elliptical orbit with the sun in focus. Using M = dL/dt, prove that a planet cannot leave its plane of orbit. Note: M here is an externally applied torque that the sun exerts on the planet. diagram of the situation described b)...

is there a rigorous version of this proof of fundamental theorem of calculus?if yes,what is it?and who came up with it? i sort of knew this short proof of the fundamental theorem of calculus since a long while...but never actually saw it anywhere in books or any name associated with it. i know...

Let a 3 × 3 matrix A be such that for any vector of a column v ∈ R3 the vectors Av and v are orthogonal. Prove that At + A = 0, where At is the transposed matrix.

Hello everyone. First off, I'm sorry if this post is excessively long, but after tackling this for so many hours I've decided I could use some help, and I need to show everything I did to express exactly what I wish to do. Also, to be clear, this post deals with integration by substitution. Now...

Consider the equivalence: (∀v Fv -> p) <=> (∃u Fu -> p) Where variable v occurs free in Fv at all and only those places that u occurs free in Fu, and p is a proposition containing no free occurences of variable v. Can someone please offer a proof of such equivalence. Many thanks. am

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So, I know it can be proven using calculus, but I need the geometric one. So, I got that ^c=^d and therefor, the amount of increment in one of a, is equal to the other(^e=^b). (Also 0<a+b<Pi/2) And AP'=BP'=BD/sin(a) and BP=BD/sin(a+b) and AP=BD/sin(a-b). AP'+BP'=2AP'=2BD/sin(a) and...

For my base case I just used a graph with three vertices and 2 edges. Decomposing this would just give us the same graph, which has a path length of 2. The inductive step is where I'm having some trouble: One idea I have is that we take a graph G then inductively remove an edge to create two...

How can you prove that ##f(x)=g(x) \Leftrightarrow f(x)+C=g(x)+C##

Dear friends, I was wondering if someone can explain how Cantors diagonal proof works. This is my problem with it. He says that through it he finds members of an infinite set that are not in another. However, 2 and 4 are not odd numbers, but all the odd numbers equal all the whole numbers. If...

Problem Statement: Prove that |a|=|-a| Relevant Equations: ##|a|= a, ## if ## a \geq 0 ## and -a, if ## a \leq 0 ## Somewhat stumped on where to start... i know that we need to use cases. If we consider ##a\geq 0##, then are we allowed to use the fact that ##|-a|=|-1|\cdot|a| = |a| ##? This...

Note sure if this belongs in the Basic Math category or Calc & Beyond section. I want to make sure I am on the right track here. Here is what i have so far: x^2 = y^2 Multiply both sides by x^-1 twice (invoking P7) x^2 \cdot x^{-1} = y^2 \cdot x^{-1} x \cdot x^{-1} = y^2 \cdot x^{-1}...

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help to fully understand the proof of...

My question concerns the portion of the proof stating, “...we set up a correspondence between the elements of U(A_n), for n in N, and a subset of S by making the element a correspond to (m, n) if A_m is the first set in which a appears, and a is the nth element of A_m.” In particular, I am...

If someone can straighten out my logic or concur with the presence of a mistake in the proof (even though the conclusion is correct, of course), I would be much obliged. I’m looking at the proof of the corollary near the middle of the page (image of page attached below). I simply don’t find...

I actually don't know how to proceed. I tried something like this The left side of the equation equals to $$\delta(\int_a^b F(x)dx)=\delta f(x) |_{a}^{b}$$ where ##f'(x)=F(x)## However $$\delta f(x) |_{a}^{b}=f'(x)\delta x dx|_{a}^{b} = \delta (F(b)-F(a))$$ where ##f'(x)=F(x)##. For the...

I have some questions about J. S. Bell’s famous theorem as presented in his1964 paper.1 These are about his theoretical assumptions and reasoning, not about experimental observations such as Aspect-type experiments. While some questions relate to the experiments, others do not because Aspect’s...

How can the inequality ##cosx \ge(1-x^2/2)## be proved? Would you please explain how to prove this inequality? This is the only equation that I could think of. ##1\ge cosx \ge 0## but I cannot use it here. Source: Thomas's Calculus, this is from an integration question there. Thank you.

Hi everyone, while I was digging arima model I saw that BIC value is given as $k*log(n)-2*log(L)$ where $L$ is the maximized value of likelihood function whereas $k$ is number of parameters. I have found the proof of AIC but no any clue about that. I wonder how it is derived. Could you help me...

Preface After a lengthy discussion of the thermal interpretation of quantum physics in https://www.physicsforums.com/threads/the-thermal-interpretation-of-quantum-physics.967116/ , now I think I can prove that it is wrong, i.e. that it doesn't solve the measurement problem in a way it claims it...

So, I found these statements and I need your assistance to prove them since my body condition is not fit enough to think that much. 1. The quadratic equation whose roots are k less than the roots of $$ax^2+bx+c=0$$ is $$a(x+k)^2+b(x+k)+c=0$$. 2. The quadratic equation whose roots are k more than...

I don't understand proof of uniqueness theorem for polynomial factorization, as described in Stewart's "Galois Theory", Theorem 3.16, p. 38. "For any subfield K of C, factorization of polynomials over K into irreducible polynomials in unique up to constant factors and the order in which the...

<Moderator's note: Moved from a technical forum.> Hi PF, I am learning how to prove things (I have minimal background in math). Would the following proof be considered valid and rigorous? If not any pointers or tips would be much appreciated! Problem: Prove that the notion of number of...

https://en.wikipedia.org/wiki/Area_of_a_circle#Onion_proof I understand the basic concept, although it is a little difficult to visualize the thin discs close to the centre of the circle. Regarding the area of each disc, it is given in the link above as 2πrdr. Then, by means of integration...

Hi, I am self studying induction and came across the following problem. I am stuck on how to proceed (I need to use induction, I know there is a direct proof). My proof attempt is as follows: Let ## P (m) ## be the proposition that: $$ \sum_{i = m + 1}^{n} i = \frac{(n - m)(n + m + 1)}{2} $$...

gcd(f_n,f_{n-1}) gcd[f_{n-1},f_n - f_{n-1}] gcd[(f_n - f_{n-1}), (f_{n-2} - f_{n-1})] gcd[(f_{n-2} - f_{n-1}),f_{n-3} - f_{n-2})] gcd[(f_{n-2} - f_{n-3}),(f_{n-4} - f_{n-3})] . . . gcd(f_2,f_1), where f_2 = 1, f_1 = 1 I assume LateX is not working yet. Not sure if I am on point here or not...

$n$ is a positive integer with the following property: If the last three digits of $n$ are removed, $\sqrt[3]{n}$ remains. Find with proof $n$. Source: Nordic Math. Contest

I can not understand why ##v_x = -|v|sin(θ)## and ##v_y = |v|cos(θ)## I'm asking about the θ angle. If i move the vector v with my mind to the origin i get that the angle between x'x and the vector in anti clock wise, it's 90+θ not just θ. So why is he using just θ? Does the minus in v_x somehow...

Sorry for the misspelling, but this forum doesn't allow enough characters for the title. The title should be: For the topological proof of the Fundamental Theorem of Algebra, what is the deal when the roots are at the same magnitude, either at different complex angles, or repeated roots? I...

Greg Bernhardt submitted a new blog post A Pure Hamiltonian Proof of the Maupertuis Principle Continue reading the Original Blog Post.